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Many of us have read that the spectrum of prime distribution will give zeros of Riemann Zeta function. For example, Mazur and Stein's book: (http://wstein.org/rh/rh.pdf ) have many nice pictures that show the spectrum of prime distribution indeed "contains" the information of zeta zeros.

But I had not seen a rigorous proof of this. I am looking for such a proof. Be specific, I am looking for a proof which shows that the peaks of the spectrum of prime distribution are the locations of zeta zeros.

Maybe such a proof exist, can anyone share a link or give some reference on such a proof ?

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    $\begingroup$ $$\sum_{n=1}^\infty a(n) n^{-s} = s\int_1^\infty A(x) x^{-s-1} dx = s \int_0^\infty A(e^u) e^{-su} du$$ where the last is the Laplace transform of $A(e^u) = \sum_{n \le e^u} a(n)$, equivalently the Fourier transform of $A(e^u) e^{-\sigma u}$ or if you prefer distributions of $\sum_{n=1}^\infty a(n) n^{-\sigma}\delta(u - \ln n)$ $\endgroup$ – reuns Jan 13 '16 at 23:04
  • $\begingroup$ and if you like the residue theorem, math.harvard.edu/archive/213b_spring_05/… (page 10 they get a proof of the explicit formula you are looking for) $\endgroup$ – reuns Jan 13 '16 at 23:28
  • $\begingroup$ The original question did not ask explicit formula. It asked about a deep question related to the "spectrum view of primes". The question is, if the spectrum of primes give zeros of zeta, then why it only gives the positions of zeros on critical line, why it does NOT give the location of critical line ? $\endgroup$ – david Jan 14 '16 at 17:30
  • $\begingroup$ what do you call the sprectrum of prime distribution ? $\sum p^{-s}$ which is the Laplace transform of $\sum_p \delta(x- \ln p)$ ? did you realize that $\ln \zeta(s)$ is the Laplace transform of $\sum_{p,k} \frac1{k}\delta(x- k \ln p)$ which is $\sum p^{-s} + \mathcal{O}(1)$ for $\Re(s) > 1/2$ ? so I gave you the hints for proving the link between the spectrum of prime localization and $\zeta$, if you don't want to look at it, it's your problem. $\endgroup$ – reuns Jan 15 '16 at 7:39
  • $\begingroup$ Did you see endnote 19 in the book that explains further and has some more references? Some of the poles of the series are "oscillatory." $\endgroup$ – Joe Knapp Feb 1 at 0:14

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